This grant proposal is targeted at research problems arising in the early detection of cancer with special emphasis on the early diagnosis of breast cancer. The principal research areas include:(i) Develop a stochastic model for the disease progression of breast ductal carcinoma in situ (DCIS), (ii) Investigate the natural history of breast cancer, including methods for estimating the pre-clinical sojourn time and lead time distributions and (iii) Develop a new method for investigating the over diagnosis of disease. The diagnosis of breast DCIS in the U.S. has risen rapidly consistent with the dissemination of mammographic screening. Little is known about the natural history of DCIS. However women diagnosed with DCIS are known to be at elevated risk for invasive breast cancer (IBC). Knowledge on the natural history of DCIS, i.e. how it develops, if and when it will progress to IBC, would be an essential first step to the management of DCIS. Modeling the natural history of DCIS is challenging as there are no ideal data on DCIS progression of untreated DCIS cases. The first main aim of this research is to develop a stochastic model to evaluate the DCIS progression process. This model will utilize the U.S. Surveillance Epidemiology and End Results (SEER) and Norwegian data. Another aim involves a theoretical investigation of the sojourn time distribution in the early stage of breast cancer and the lead time associated with early detection of cancer. The last aim develops a new method for over diagnosis estimation. It arises in the early detection of disease setting, by screening programs finding cases which would never have become clinical in one's life time. Currently the actual level of over diagnosis from breast cancer screening is in dispute. It is planned to develop a new statistical method for estimating over diagnosis. This method will be unique in that it will make use of the disease incidence data. Applications will be made using the data from the Norwegian Breast Cancer Screening Program. Overall the proposed research will contribute greatly to the management of DCIS in clinical practice and shed some light on ongoing debate of 'harms of breast cancer screening.